The article is extremely interesting, and the aim of this post is to illustrate what was done in the article with some nice graphs (with some codes written by Mohamed Amine, student in my MAT8595 course) to visualize inference aspects. As in ultimate 100m world records through extreme-value theory, consider the fastest personal best times that were set in a certain period (each athlete only appears once on the dataset). As in the paper, consider observation after 01-01-1991, until 19-06-2008. One can update the dataset to take into account Usain Bolt’s 9:58 record, I’d love to see how the estimates change actually (one can also look at @tomroud‘s great post, published in 2009 on convexity issues – in French – about Usain Bolt’s records). So, here is the data,

The problem is that, in extreme value models, we usually focus on the maximum of a sample, not the minimum. A great idea is not to study tail behavior of the distribution of the times, but the distribution of the speed,

Now, to the idea is that the associated distribution is in the max-domain of attraction of Weibull distribution (and therefore, there should be some upper limit). So we need to estimate the tail index using an appropriate estimator, such as the moment based estimator introduced in Dekkers, Einmahl and de Haan (1989),

Here, we can clearly see that the tail index is negative, meaning that the distribution of the speeds is in the max-domain of attraction of the Weibull distribution, and thus, the support of the distribution has an upper bound.

But if we want to get a proper estimation of the tail index, which should we consider? As suggested again in the original paper, consider the idea of Beirlant, Dierckx & Guillou (2005), to derive an optimal ,